Diagrams and their role in mathematical proof

Diagrams in mathematical proofs are typically used only for the purposes of explanation and to aid intuition: they are not seen as formal objects which constitute part of the proof itself.
The two main justifications for this attitude are that the use of diagrams can lead to errors in reasoning, and that diagrams are expressively weak.
We examine these points of view to determine whether they are valid or not, and how they can be overcome.
The examples used will be from the domains of logic and formal language theory, and will be based upon Venn and Euler diagrams.
In particular, we will describe how any regular language can be encoded in an extension of an Euler diagram, known as a spider diagram.